Who said differential dx is a very small quantity? Given a differentiable function $f(x)$ in a point $x_0$, by definition, the differential $df$ is the difference in ordinates of the tangent line to curve in $x_0$, evaluated at the point $x_0 + dx$.
Now, $dx$ can be small or big, and if it is small enough, we know that this difference approximates: $f(x_0 + dx) - f(x_0)$.
Question is: it's $dx$ just a quantity which can be taken big or small? (often taken small enough in applications)?
 A: It depends on the convention used in the book you are consulting. As an example, when you write $\int f(x) dx$, then the $dx$ does not have a proper meaning in itself, because one reads it all together with the notation $\int ... dx$. However, it is heuristically understood to be meaning infinitesimal portion along the $x$-axis.
In differential geometry, $dx$ is often used to indicate, for instance, the basis vector to the tangent plane to the manifold. Since such a plane will have a different basis for each infinitesimal movement along the surface of the manifold, geometrists and physicists decided to denote it like that, because it heuristically makes sense.
To sum up: it depends a lot on the authors of the books and what they want to convey with that notation.
A: This is not going to be rigorous, but in a way, $dx$ can be represented as a quantity, but I like to see it as more of a change in the linear approximation using a tangent line on a curve. I wouldn't think of $dx$ as a "big quantity." It's more like a small $\Delta x$ that, when approaching $0$, gets closer to $dx$.
Note that $\Delta x$ is the difference between $x$-values and the original $x$-value. But $dx$ is the differential and it's an approximated difference between the $x$-values. For functions $y = f(x)$, the differential $dy$ is just an approximated change in the $y$-values by using the derivative.
In a standard calculus class, it's fine to think of $dx$ and $dy$ as infinitesimal quantities if you don't mind being informal and if you want to get a basic intuition.
